11.3: Sexuality in Early Adulthood
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Learning Objectives
By the end of this section, you will be able to:
- Define young adult sexuality
- Define sexual identity and orientation
- Explain the connections between sexual activity and health
- Explain the risk factors for and consequences of sexual assault and harassment
During adolescence, David realized he was attracted to other males.  Away at college in a large urban center, David joined the school’s LGBTQ+ student group and was fully out to his university community; however, when he went home on breaks, he “went back in the closet.”  As their connection grew, it became more difficult, and more uncomfortable, to hide this important part of his life. After some minor mistruths about how holidays and breaks were being spent, David decided it was time to come out to his family.  With their support, David was able to gradually start living as his authentic self in his hometown. Several years later, he is closer to his family than ever.
Emerging adulthood is typically a time of sexual exploration along with exploration of other aspects of ourselves. This can include the formation of social and emotional bonds with others, which we’ll discuss in more detail in Chapter 12 Social and Emotional Development in Early Adulthood (Ages 18 to 29). However, sexuality also has physical and cognitive aspects, such as achieving sexual maturity and choices about sexual behavior. Here, we’ll focus on these dimensions of sexuality as a part of adult development.
Sexual Maturity and Sexual Behavior
The Sexuality Education Resource Centre (SERC) in Manitoba, Canada, which states that a person’s sexuality is self-defined and can change over time depending on age and circumstances. The SERC portrays sexuality as being in a “wheel” composed of different components that interact with each other (SERC MB, n.d.), as seen in Figure 11.13.
Sexuality is a complicated concept with many influences, including biology, culture, religion, the views of our family and friends, and the portrayals of sexuality in mass media (Macleod & McCabe, 2020; Potard et al., 2008) (Figure 11.14). Because culture evolves, so do attitudes and behaviors regarding sexuality. For several decades, the General Social Survey (GSS) from the University of Chicago has tracked U.S. societal opinions on attitudes toward various facets of sexuality. Over that time studies have found greater acceptance toward living together before marriage and having sex before marriage (GSS Data Explorer, 2022; Harding & Jencks, 2003). However, in the United States and some European countries, these behaviors are now widely accepted, though they are still frowned upon or even prohibited in other countries (Rokach & Patel, 2021). Societal acceptance of non-heterosexual relationships, and the experiences people have when coming out, likewise vary by cohort (Martos et al., 2015; van Bergen et al., 2021). In some parts of the world, young females undergo procedures meant to decrease sexual pleasure, and marriage sometimes occurs before reaching puberty to reduce the incidence of premarital sex (Rokach & Patel, 2021; UN Women, 2019). However, in other cultures, premarital sex is considered a natural aspect of the transition from adolescent to adult and is encouraged (Christensen, 1960; Rokach & Patel, 2021).
Why do people engage in sexual activity? Beyond the obvious reasons that it’s enjoyable, there are psychosocial reasons as well. Sexual activity is a way to express feelings to and establish connections with others, an important developmental task of emerging adulthood.  Social factors may also influence this (Figure 11.15). A study examining heterosexual women’s same-sex encounters—specifically, kissing publicly—at college parties indicated that there was no single motive for this behavior; instead, it was motivated by one of several factors, such as being drunk, wanting to experiment, and enjoying the attention this behavior produced, especially attention from men (Yost & McCarthy, 2012). Sexual activity is also considered to be a standard element of romantic relationships, and some people engage in it out of feelings of obligation or duty—feelings that can be exploited, particularly in cultures where women are expected to be submissive and marital rape isn’t regarded as a crime (UN Women, 2019).
The experience and expression of sexuality are unique to everyone. The peak of sexual activity generally occurs between the ages of 18 and 24, though the number of adults in this age group who were sexually active decreased somewhat between the years 2000 and 2018, especially men (Ueda et al., 2020). Becoming sexually active is a common part of growing up, although it’s not a mandatory element (i.e., people who are asexual or who abstain from sexual activity for religious or other reasons aren’t somehow “defective”). Sexual activity is also, for many people, a healthy and enjoyable aspect of their lives. However, sometimes sexual activity has unwanted health consequences, such as sexually transmitted infections or unintended pregnancy.
Sexual Orientation and Gender Identity
Recent data indicate that in the United States, about 7 percent of the population identifies as LGBTQ+, including 20 percent of adults in “Generation Z”, the group born between 1997–2012 (Jones, 2024). Looking beyond the U.S., a survey of 30 countries indicated that about 9 percent of the world’s population identifies as LGBTQ+, although there’s a lot of variability between countries; for example, 15 percent of Brazil’s population identifies as LBGTQ+, but only 4 percent of Peru’s population does (Jackson, 2023). In both sets of data, the most common LGBTQ+ identity was as bisexual; about 4 percent of the U.S. and world population identify as bisexual, and 57 percent of the LGBTQ+ adults in the U.S. claim this identity (Jackson, 2023; Jones, 2024).
 In cultures where LGBTQ+ people are stigmatized or even criminalized, there may be significant risks to coming out, and some people may choose not to do so until later in life if at all (Rosati et al., 2020). Even in countries with LGBTQ+ friendly policies, that doesn’t mean that individual people or communities will be accepting.  These tasks typically don’t occur simultaneously, although they do appear to happen in a fairly consistent order, and the specific timing of these tasks may vary. In general, men achieve these tasks earlier than women, and people identifying as gay or lesbian achieve most of these tasks earlier than people identifying as bisexual. There are also cohort differences in that people ages 45–59 reported the relationship and disclosure tasks occurring later than people ages 30–44, who in turn reported these tasks occurring later than people ages 18–29. These differences likely reflect changing societal attitudes regarding the acceptability of non-heterosexual relationships (Martos et al., 2015; van Bergen et al., 2021).
The study by Martos and colleagues indicates that identifying as lesbian, gay, or bisexual commonly happens in mid-adolescence, but disclosure to others may not occur until emerging adulthood (2015). Emerging adulthood, as we’ve already seen, is a time of continued identity exploration and commitment after we’ve gained more independence. This pulling away from our family is often accompanied by increased connections with similar peers. Studies of college students in the United States suggest that between 7 percent and 14 percent of heterosexual women and roughly 4 percent of men report having at least one same-sex encounter. These rates are lower in other countries, such as Thailand, most likely due to cultural differences in the acceptability of same-sex involvement (Morgan, 2013). Heterosexual women consistently report more same-sex attraction and encounters than heterosexual men do, potentially because of internalized standards of “appropriate” male behavior and a general greater incidence of sexual fluidity in women (Morgan, 2013; Yost & McCarthy, 2012) (Figure 11.16).
Sexually Transmitted Infections
Because sexual activity generally peaks during this time, it’s not surprising that the years from ages 18–24 are also the peak years of risk for contracting a sexually transmitted infection (STI) , a disease caused by certain types of viruses, bacteria, or microorganisms and spread through contact with bodily fluids such as semen and blood. According to the Centers for Disease Control and Prevention (CDC), in 2022 almost half the new cases of STIs occurred in people between 15 and 24 years of age (CDC, 2024). The use of the phrase “sexually transmitted” is a little misleading, because some STIs can be acquired outside of sexual activity; for example, HIV can be passed to nursing babies through breastmilk. However, regardless of the mode of transmission, the health effects of STIs are the same.
There are several different STIs, each with its own cause. Syphilis, chlamydia, and gonorrhea are caused by bacteria. Trichomoniasis is caused by a parasite. Common symptoms of these four STIs are painful urination, discharge (including blood) from the genitals or anus, and pelvic or abdominal pain, although some people may not have any symptoms. These STIs can be treated with antibiotics; however, a person can get re-infected if they come into contact with the bacteria/parasite again. It’s also important to take the antibiotics as directed, including taking all the medicine even if symptoms improve.
Other STIs are caused by viruses. Genital warts are caused by the human papillomavirus (HPV) and typically occur as small cauliflower-like bumps on the genitals, although some people may not have any symptoms. These warts can be treated with special medicine (not the same over-the-counter medicine used to treat warts on other parts of the body like the fingers). Herpes is caused by the virus herpes simplex , which comes in two forms. Herpes simplex virus type 1 (HSV-1) typically causes “oral herpes” and most often appears as cold sores around the mouth. HSV-1 isn’t technically an STI, although it can be contracted from a person with HSV-1 through kissing or oral sex.  Herpes simplex virus type 2 (HSV-2), on the other hand, is usually spread just through sexual activity, although it can also be passed to babies during a vaginal birth. It typically appears as blisters or sores on the genitals, although again some people may not have any symptoms. The blisters may go away after treatment with medication, but may come back during times of illness or stress. When we talk about someone “having herpes”, we’re most often talking about HSV-2, though not always.
The human immunodeficiency virus (HIV) produces different symptoms than HPV and HSV-2. The initial infection typically involves not warts or sores on the genitals, but flu-like symptoms such as fever, chills, fatigue, and a sore throat. Often these symptoms go away within a couple of weeks, and then the person may not have any symptoms for as many as 10–15 years (HIV.gov, 2022). Unfortunately, this means that people may unknowingly spread HIV to others during that time. Also, without treatment, the virus slowly destroys the body’s immune system, and the person eventually develops severe medical problems such as pneumonia, tuberculosis, dementia, and some types of cancers. Some of these are opportunistic infections, medical conditions that occur more frequently in people with weakened immune systems who can’t fight them off. An HIV-positive person who develops these types of medical problems is now said to have acquired immunodeficiency syndrome (AIDS), a late stage of HIV infection in which the body’s immune system is significantly weakened and susceptible to a wide variety of diseases.
A viral STI can’t be treated with antibiotics, and a person with HPV or HSV-2 will carry the virus for the rest of their life. They may need to take medication regularly to prevent or treat flare-ups. Similarly, a person with HIV will need medication lifelong to prevent the virus from continuing to multiply inside their body. With early and high-quality medical care, they can enjoy good health and quality of life for many years.
The incidence of STIs is unequal across the population. Non-Hispanic Black individuals make up only 12.6 percent of the U.S. population but account for 31 percent of chlamydia, gonorrhea, and syphilis cases. This high proportion is likely due not to a greater incidence of sexual behavior, but to lack of access to reproductive and sexual health care (CDC, 2024), typical for marginalized groups. For example, HIV is more common among men who have sex with men (CDC, 2021); in the early days of the HIV pandemic, the disease wasn’t taken seriously and people with HIV were treated poorly because both being gay and having a disease were stigmatized.
Diagnosing and treating STIs is crucial to avoid permanent health consequences.  HPV can’t be eliminated from the body, but a person who knows they have it can get regular medical checkups to detect cancer early enough for treatment. Chlamydia, gonorrhea, and syphilis can all cause infertility, chronic pelvic pain, and complications in pregnancy such as premature birth. If untreated for decades, syphilis can damage the brain and produce blood clots, stroke, and dementia. Treatment for HIV works best if started early, and while people who are HIV-positive may not know for several years, testing is available and recommended.
The stigma associated with STIs—and sometimes with sexual activity in general—can prevent people from seeking treatment. Deciding to engage in sexual activity requires taking increased responsibility for monitoring your health, however, including getting regular STI testing and being honest with health care providers about behaviors and symptoms.
In recent years, treatments for HIV have dramatically improved meaning that most people with HIV can manage the virus through antiretroviral therapy (ART). ART involves taking medicines that allow people to get and stay at a level of virus suppression that helps them to stay healthy and means they will not transmit HIV to other during sexual intercourse (CDC, 2024).
Pregnancy in Early Adulthood
You learned earlier about the process and milestones of pregnancy and childbirth (Chapter 2 Genetic, Prenatal, and Perinatal Health). Pregnancy is exciting and welcome news for many (Figure 11.11), but not for all. “Unintended pregnancy” refers to any situation in which a person becomes pregnant without intending to.
In the United States, approximately 3.6 million people give birth each year, with 48.6 percent of these people being between the ages of 18–29 (Osterman et al., 2024). Determining how many of these pregnancies are unintended is challenging because many of them are likely to occur among people who aren’t married, and being a single parent is still associated with some stigma in many cultures. Therefore, people may not report this information thoroughly.
As you learned in Chapter 2 Genetic, Prenatal, and Perinatal Health, pregnancy causes many physical changes, which can be hard on the body during the pregnancy and postpartum period. Worldwide, maternal mortality is still a concern, with around 287,000 birth mothers dying in childbirth each year—most often in middle and lower income countries (World Health Organization, 2024). However, most pregnancy-related deaths can be prevented through effective prenatal and perinatal care (CDC, 2024) and the birth mother's body generally physically recovers from the pregnancy and delivery during the first couple months after delivery (Lopez-Gonzalez & Kopparapu, 2022). For those who decide to keep and raise a baby, becoming a parent is a significant transition that's expensive, emotionally challenging but often rewarding, and. time-consuming.
This is likely one of the reasons, we see a societal trend to delay parenthood until later in early adulthood or even in middle adulthood for those who choose to become parents. In addition, we have seen trends of more people choosing not to become parents often citing financial concerns (APA, 2024). Mothers in particular still take on a higher caregiving load in most families with small children and additionally face a high deal of societal pressure, often known as "mommy guilt" to bear the main responsibility of the child's wellbeing (Constantinou et al., 2020). This pressure can be present across family types including in particular single mothers and heterosexual couples, even when the couple has a high level of shared responsibility. While fathers take on much more caregiving responsibilities than was common in decades past in the U.S., mothers are still much more likely to reduce or leave the workforce than fathers (Heggeness, 2020).
Choosing to become a caregiver and starting a family can be a fulfilling part of an individuals identity. Planned parenting often brings psychological fulfillment and wellbeing, enjoyment of family socioemotional experiences, and can contribute to a person's sense of meaning in life (Chen et al., 2016). This can mean that for many becoming a parent can be another important part of identity formation and commitment, best facilitated by having social support, control over the choice to become a parent, and knowledge about parenting effectively (Chen et al., 2016; Piotrowski, 2020).
Parenthood can also affect the transition to adulthood. Becoming a parent is a very adult, and often rewarding, responsibility that can help define our identity. But in early adulthood it can also interfere with the achievement of identity tasks such as furthering an educational or training program and starting a career (Landberg et al., 2019) (Landberg et al., 2019). People may find an identity in becoming a parent, but it may also delay or prevent them from exploring other facets of their identity; of course, we can say the same about other choices we commit to.
Both STI and pregnancy prevention share some similar strategies, and some foundational elements. First, without completely avoiding sexual activity (known as abstinence ), guaranteed prevention of STIs and pregnancy is nearly impossible. Thus, there is no truly “safe sex” and precautions instead can provide “safer sex.” When choosing a pregnancy or STI prevention method, it’s important to be aware of your health status and use good decision-making to determine what’s likely to work best for you and any partners.
Abstinence is often said to be the most reliable method for avoiding STIs and pregnancy (CDC, 2023). However, not everyone understands “abstinence” the same way. Some people consider abstinence to mean avoiding any kind of sexually-related contact, including oral sex and mutual masturbation. Others use this term only to mean avoiding penetrative intercourse (Byers et al., 2009; Hans & Kimberly, 2011). These varying definitions of this term can give different impressions about a person’s sexual activity and could put them at risk for contracting an STI. For example, a person who engages in oral sex but not intercourse could contract herpes while still considering themselves abstinent. When discussing abstinence, either in a personal conversation, a health-related setting, or a research context, it’s important to make sure you know how others define abstinence.
Other behavior-based STI prevention methods include being in a monogamous relationship, reducing one’s number of sex partners, and avoiding drug and alcohol use. Because drugs and alcohol can impair judgment, people who use substances are at higher risk of STIs because they may engage more often in risky sexual behaviors like unprotected sex or sex with multiple partners (Baskin-Sommers & Sommers, 2006; Kusunoki & Barber, 2020). There are also medications (both shots and pills) to reduce the chances of getting HIV. These medications are called pre-exposure prophylaxis (PrEP) and are meant to be taken on a regular basis—a similar strategy as birth control pills, but for preventing HIV and not pregnancy.
Apart from abstinence, condoms are the only method effective at preventing both pregnancy and STIs. Condoms are called a barrier method because they prevent exchange of bodily fluids. Another barrier method used for STI prevention is a dental dam , a thin piece of latex or polyurethane that’s placed over the vulva or anus prior to oral sex. Other barrier methods that are used as pregnancy prevention but not STI prevention include diaphragms, cervical caps, and contraceptive sponges. Each of these is inserted into the vagina prior to sex and work to keep semen from reaching the uterus. They’re also typically used with spermicide , a gel that kills sperm cells, increasing the effectiveness of pregnancy prevention.  Removal should be done following the product’s directions or as advised by a health-care provider to optimize their effectiveness.
Some pregnancy prevention methods are hormone based. Birth control pills, contraceptive shots, and contraceptive implants each contain the synthetic hormone progestin, which prevents ovulation. Contraceptive shots and implants are long-lasting (3 months and 5 years respectively). Birth control pills must be taken daily, and medications like antibiotics can have contraindications that may make them ineffective.
Other methods of pregnancy prevention are more complex. An intrauterine device (IUD) is a small T-shaped device that’s inserted into the uterus via the vagina. It works by either preventing ovulation or reducing the movements of sperm. Unlike a diaphragm, an IUD can stay in place long-term (several years); however, it must be inserted and removed by a health-care provider. There are also surgical birth control methods.
Link to Learning
Many college campuses as well as local community health clinics may have health centers where you can get tested for STIs. However, if you’re not able to access this service or choose not to, there are tools you can use to find a testing center.
- If you’re in the United States, you can find STI testing locations through the CDC website.
- If you’re in Europe, you can use the European Test Finder website to search by country, city, and type of services needed.
- If you’re in Africa, South America, or Asia, you can use the Building Healthy Online Communities site.
Sexual Harassment and Sexual Violence
So far we've been talking about sexual activity as always consensual, relatively common, and hopefully enjoyable aspects of life in early adulthood.
 Force can involve verbal, physical, or emotional behaviors, such as manipulation. Sexual violence may include assault, abuse, harassment, exploitation, public exposure, or stalking behaviors. Regardless of the type of sexual violence, all acts of sexual violence involve harm to another person. Sexual harassment is a broad term referring to a range of actions representing unwanted physical or verbal attention with a sexual basis. It most commonly involves behaviors that make a person uncomfortable and interfere with that person’s ability to engage in typical activities such as going to school, working, or participating in extracurricular settings, like a sports club or religious group.  Repeatedly making negative comments about a specific gender or talking about sexual acts or fantasies could constitute sexual harassment even if it just occurred within earshot of a person who wasn’t directly included in the conversation. Sexual harassment often involves a power differential in that the harasser may be in a position of authority, such as a boss, teacher, or spiritual leader. However, this isn’t a requirement to define sexual harassment; it can occur among peers, such as among teammates or classmates.   Consent must be explicitly given; “not saying no” is not the same as saying “yes”. A person who’s unconscious or too scared to refuse has not given consent . Neither has a person who’s unable to make informed decisions due to intoxication or cognitive impairment.
Link to Learning
If you or someone you know are experiencing sexual assault or harassment, reach out to the Rape, Abuse & Incest National Network (RAINN) for help. Call the National Sexual Assault Hotline at 1-800-656-HOPE or chat through the RAINN website for immediate, confidential support.
 According to a 2019 survey of 893 adults in the United States, nearly a third of these participants (32.7 percent) reported experiencing sexual harassment between the ages of 18–29, with 24.6 percent reporting experiencing sexual assault.  For context, the rates of sexual harassment in this age group were 105 percent higher than for respondents over age 30, and the rates of sexual assault were 65 percent higher. Women were more likely than men to report sexual harassment (37.4 percent vs 22.4 percent) and sexual assault (36.0 percent vs 16.0 percent; Mumford et al., 2020). These rates in general are also comparable to many other countries; statistics indicate that approximately a third of women worldwide experience sexual violence (UN Women, 2019).
Unfortunately, college campuses—common environments for young adults—aren’t immune to these problems. A large-scale study of sexual assault and harassment reports on college campuses between 1966–2017 found that between 11–73 percent (median 49 percent) of women reported sexual harassment at college. Sexual harassment was more likely to happen to women who were younger, economically insecure, and of an ethnic or sexual minority group. All of these factors indicate a vulnerable population, and perhaps not surprisingly, this study noted that over half these incidents had gone unreported to university personnel. Interestingly, data indicated rates of sexual assault and violence declined during the period being examined, While sexual violence is a higher risk for women and girls, research estimates that 27 percent of men have been victimized at some point in their lives (Thomas & Kopel, 2023). Additionally, transgender individuals are at a much higher risk of being victims of sexual violence (Stotzer, 2009). Due to the prevalence of these crimes, prevention and intervention work needs to include a community and societal wide response.
Preventing Sexual Violence and Sexual Harassment
One issue sometimes noted is that many “prevention efforts” are aimed at potential victims, not at potential perpetrators. For example, telling people to keep an eye on their drinks at parties is helpful, but it would be even more helpful if other people didn’t slip drugs into others’ drinks in the first place. Suggesting that women not walk alone at night is another example that focuses the burden of preventing sexual violence on women and unintentionally may contribute to victim blaming and other rape myths. Higher endorsement of rape myths, including those that put the burden on potential victims, is associated with an increased likelihood of perpetrating sexual assault as well as with a higher rate of sexual assault in that community - including on college campuses (O'Connor, 2021). At the root of many cases of sexual- and gender-based discrimination appear to be individual and cultural beliefs about gender roles and behavior; e.g., men are expected to be dominant while women are expected to be submissive, even regarding consenting to sex (UN Women, 2019). These beliefs are common in many African and Asian cultures (de Villiers et al., 2021; Olson-Strom & Rao, 2020) and is also reflected in the Latino values of machismo (Kim et al., 2017).  Teaching and practicing safety behaviors is important, but a more proactive approach would involve addressing the reasons why people mistreat others in the first place.
Some groups are attempting to do just that. For example, Men as Peacemakers, a nonprofit organization located in Minnesota, provides programming aimed at teaching men to avoid and resist all forms of violence against women, including some programming aimed specifically at college students. Similarly, in South Africa, the One Man Can Intervention addresses specific cultural views of gender norms and encourages men to become advocates for gender equality in all forms, including ending sexual violence. While data regarding the effectiveness of these types of programs are somewhat mixed, the results generally lend support to the idea that both attitudes and behaviors can be changed and thus lead to lower rates of discrimination and maltreatment (de Villiers et al., 2021; Flood, 2022; Wright et al., 2020).
The anti-sexual assault movement began in the U.S. with major changes in legal definitions shifting up through the 1990s. For example, marital rape was not recognized as a crime in every U.S. state until 1993 and the Violence Against Women Act was pivotal in shaping current policies and advocacy resources beginning in 1994 (WCSAP, 2024; Tulane University, 2024). This was followed by the establishment of many organizations and a wealth of advocacy work, often performed by women and survivors, to both prevention sexual violence and support recovery for survivors of sexual violence (Tulane University, 2024).
Other changes occur at a public policy level. In the United States, for example, Title IX is a federal policy aimed at addressing sexual and gender-based discrimination in educational settings. It contains regulations for reporting and responding to incidents of sexual assault, abuse, and harassment, including provision of counseling services to survivors, even if these incidents didn’t involve the school environment. However, changes were made during the Trump administration due to concerns about lack of due process for people accused of sexual violence. Following these changes, sexual violence on college campuses increased by 3 percent for women and 1.5 percent for men (Rochester, 2022). Tavares and Wodon (2018) theorized that higher education was weaker than the workplace in addressing sexual harassment, noting that as of 2017, 60 percent of countries worldwide lacked laws addressing sexual harassment in education settings, while only 21 percent lacked these laws for the workplace. Finally, sexual violence can be prevented by focusing more on improved early sexual education and a better understanding of consent culture, to learn more refer to 9.2 Puberty, Sexual Behavior, and Sexual Health in Adolescence.
Intersections and Contexts: Title IX and Equivalents around the World
Title IX was first enacted in the United States in 1972 to address inequities in education (Rochester, 2022), but has since been expanded to include all forms of discrimination based on sex and gender. Under this policy, any educational institutions receiving federal money are required to provide a fair educational experience to their students regardless of sex, gender, sexual orientation, or gender identity. This includes avoiding gender-based discrimination and preferential treatment in the classroom, allocating equivalent resources to athletics teams regardless of player gender, and responding quickly and appropriately to incidents of sexual harassment, sexual abuse, and sexual assault.
Do countries outside the U.S. have similar policies to protect students? It depends on where you go. In 2011, the human rights organization Council of Europe (CoE) developed a treaty called the “Istanbul Convention”, which states that governments have a responsibility to prevent violence against women. This treaty has been ratified by 39 of the 46 nations that belong to the CoE, including 22 of the 27 nations in the European Union (Council of Europe, 2011), and has led to improvements in how many of those nations handle these issues. For example, in France, each public university is now required to have an employee trained to handle issues of gender inequality and discrimination, including sex- and gender-based maltreatment (Rochester, 2022).
In other places, the outlook is less optimistic, often in ways that reflect the gender-related views of the larger culture (Eng & Yang, 2020; Rao & Olson-Strom, 2020). As of 2017, 65 percent of countries in the East Asia and Pacific region and 86 percent of countries in the Middle East and North African region lacked laws addressing sexual harassment in education settings (Tavares & Wodon, 2018). In some Asian cultures, textbooks and course content downplay or even eliminate discussion of women’s contributions to the field of study, and instructors may openly show bias toward male students (Olson-Strom & Rao, 2020). LGBTQ+ students may have even more difficulty due to the stigma that exists in many cultures; for example, expressions of same-sex or same-gender sexual activity or interest are considered criminal acts in some nations, including Malaysia and Brunei. This makes it hard for students to seek help and also makes it difficult for public universities to support LGBTQ+ students due to likely backlash (Bondestam & Lundqvist, 2020; Eng & Yang, 2020).
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8.9: Sample problems and solutions
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Exercise \(\PageIndex{1}\)
A ball of mass \(m\) is dropped onto a vertical spring with spring constant \(k\) . The spring will compress until the ball comes to rest. How much will it compress if the ball is dropped from a height \(h\) above the spring?
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The two forces acting on the ball are gravity and the spring force. Both are conservative, so we can use conservation of mechanical energy. We will find the energy of the ball when it is at a height \(h\) above the spring, and the energy of the ball when the spring is fully compressed. Then, we will use conservation of mechanical energy to determine the compression of the spring.
Remember that the total mechanical energy is the sum of the total potential energy and the kinetic energy, \(E=U+K\) . Let’s call the initial position of the ball \(A\) and the final position of the ball \(B\) . You will notice that we set up our coordinate system so that \(y\) is positive upwards, with \(y=0\) at the point where the ball comes into contact with the spring. We choose to define both the gravitational potential energy and spring potential energy so that they are zero at \(y=0\) .
Since the ball starts from rest, its kinetic energy is zero at position \(A\) . At this point, the ball is not touching the spring, so the potential energy from the spring force is zero. The mechanical energy of the ball at position \(A\) is simply equal to its gravitational potential energy:
\[\begin{aligned} E_A&=U_A+K_A\\[4pt] E_A&=mgh\end{aligned}\]
At position \(B\) , the ball is again at rest, so the kinetic energy of the ball is zero. Now that the ball is in contact with the spring, it will experience a force from the spring that can be modeled with a potential energy \(U(y)=\frac{1}{2}ky_1^2\) , where \(y_1\) is the distance between the rest position of the spring and its compressed length. At point \(B\) ( \(y=-y_1\) ), the ball will have both spring and gravitational potential energy, so its mechanical energy at position \(B\) is given by:
\[\begin{aligned} E_B&=U_B+K_B=U_B\\[4pt] U_B&=mg(-y_1)+\frac{1}{2}ky_1^2\\[4pt] E_B&=-mgy_1+\frac{1}{2}ky_1^2\end{aligned}\]
Since mechanical energy is conserved in this system (no non-conservative forces are doing work), we can now set \(E_A=E_B\) and solve for \(y_1\) :
\[\begin{aligned} E_A&=E_B\\[4pt] mgh&=-mgy_1+\frac{1}{2}ky_1^2\\[4pt] 0&=\frac{1}{2}ky_1^2-mgy_1-mgh\\[4pt]\end{aligned}\]
where in the last line we rewrote the expression as a quadratic equation. We can solve for \(y_1\) with the quadratic formula:
\[\begin{aligned} y_1=\frac{mg\pm\sqrt{(mg)^2-4(1/2k)(-mgh)}}{k}\\[4pt] y_1=\frac{mg\pm\sqrt{mg(mg+2kh)}}{k}\end{aligned}\]
We now have an expression for the amount the spring is compressed, \(y_1\) , in terms of our known values.
Exercise \(\PageIndex{2}\)
A simple pendulum consists of a mass \(m\) connected to a string of length \(L\) . The pendulum is released from an angle \(\theta_0\) from the vertical. Use conservation of energy to find an expression for the velocity of the mass as a function of the angle.
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We are going to find a general expression for the energy of the system, and then use this expression to find the velocity at any point. There are two forces acting on the mass:
The force of tension (from the string). This force is perpendicular to the direction of motion at any point, so it does no work on the mass.
The force of gravity, which has a potential energy function given by \(U(y)=mgy\) . We choose the gravitational potential energy to be zero when the pendulum hangs vertically (when \(\theta=0\) and \(y=0\) ).
The mechanical energy of the mass is conserved, and at any point is given by the sum of its kinetic and its gravitational potential energies:
\[\begin{aligned} E=mgy+\frac{1}{2}mv^2\end{aligned}\]
We want to find the velocity as a function of \(\theta\) , so we need to write \(y\) in terms of \(\theta\) . As you may recall from Problem 7.6.2 , we saw that from the geometry of the problem, we can express the height of the mass as \(y=L-L\cos\theta\) , or \(L(1-\cos\theta)\) , where \(y\) is the height as measured from the bottom point of the motion. You can refer to Figure 7.6.4 to refresh your memory. The energy at any point is then:
\[\begin{aligned} E=mgL(1-\cos\theta)+\frac{1}{2}mv^2\end{aligned}\]
Conservation of energy tells us that the total energy at any point must be the same as the initial energy. So, we can use our initial conditions to find the total energy of the system. The mass starts from rest (initial kinetic energy is zero) an angle \(\theta_0\) above the vertical:
\[\begin{aligned} E&=mgL(1-\cos\theta)+\frac{1}{2}mv^2\\[4pt] E_{initial}&=mgL(1-\cos\theta_0)\end{aligned}\]
Now that we have found the total energy of the system, we can write our general expression for the energy of the system at any point:
\[\begin{aligned} E&=mgL(1-\cos\theta)+\frac{1}{2}mv^2\\[4pt] mgL(1-\cos\theta_0)&=mgL(1-\cos\theta)+\frac{1}{2}mv^2\end{aligned}\]
All that’s left to do is simplify the expression and rearrange for \(v\) :
\[\begin{aligned} mgL(1-\cos\theta_0)&=mgL(1-\cos\theta)+\frac{1}{2}mv^2\\[4pt] gL(1-\cos\theta_0)-gL(1-\cos\theta)&=\frac{1}{2}v^2\\[4pt] gL-gL\cos\theta_0-gL+gL\cos\theta&=\frac{1}{2}v^2\\[4pt] gL(\cos\theta-\cos\theta_0)&=\frac{1}{2}v^2\\[4pt] \therefore v&=\sqrt{2gl(\cos\theta-\cos\theta_0)}\end{aligned}\]
Discussion
We can see from this expression that the speed will be maximized when \(\cos\theta\) is maximized, which will occur when \(\theta=0\) (when the pendulum is vertical). This is as we expected. We can also see that we will get an imaginary number if the magnitude of \(\theta\) is greater than \(\theta_0\) , showing that the motion is constrained between \(-\theta_0\) and \(\theta_0\) . Finally, we showed that the velocity of the pendulum does not depend on the mass!
Exercise \(\PageIndex{3}\)
A block of mass \(m\) sits on a frictionless horizontal surface. It is attached to a wall by a spring with a spring constant \(k\) . The mass is pushed so as to compress the spring and then it is released (Figure \(\PageIndex{3}\)). Use the Lagrangian formalism to find an equation of motion for the mass/spring system (i.e. use the Lagrangian to determine the acceleration of the mass).
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We are going to find an equation of motion of the system using the Lagrangian method. We choose to use a one dimension coordinate system, with the \(x\) axis defined to be co-linear with the spring, positive in the direction where the spring is extended, and set the origin to be located at the rest position of the spring. The kinetic energy and potential energy of the mass are given by
\[\begin{aligned} K&=\frac{1}{2}mv_x^2\\[4pt] U&=\frac{1}{2}kx^2\end{aligned}\]
since the only force exerted on the mass that can do work is the force from the spring. We have chosen the potential energy to be zero at \(x=0\) . The Lagrangian for this system is:
\[\begin{aligned} L&=K-U\\[4pt] L&=\frac{1}{2}mv_x^2-\frac{1}{2}kx^2\end{aligned}\]
The Euler-Lagrange equation in one dimension is:
\[\begin{aligned} \frac{d}{dt}\left(\frac{\partial L}{\partial v_{x}}\right)-\frac{\partial L}{\partial x} = 0\end{aligned}\]
We can calculate the terms of the Euler-Lagrange equation:
\[\begin{aligned} \frac{\partial L}{\partial v_{x}}&=\frac{\partial}{\partial v_{x}}\left(\frac{1}{2}mv_x^2-\frac{1}{2}kx^2\right)\\[4pt] &=mv_x\\[4pt] \therefore \frac{d}{dt}\left(\frac{\partial L}{\partial v_{x}}\right)&=\frac{d}{dt}(mv_x)\\[4pt] &=ma_x\\[4pt] \textrm{and}\qquad \frac{\partial L}{\partial x}&=\left(\frac{1}{2}mv_x^2-\frac{1}{2}kx^2\right)\\[4pt] &=-kx\end{aligned}\]
and then put them together to get:
\[\begin{aligned} \frac{d}{dt}\left(\frac{\partial L}{\partial v_{x}}\right)-\frac{\partial L}{\partial x} &= 0\\[4pt] \therefore ma_x&=-kx\\[4pt]\end{aligned}\]
We can see that this equation of motion is equivalent to Newton’s Second Law.


4.3: Operations on Limits. Rational Functions
I. A function \(f : A \rightarrow T\) is said to be real if its range \(D_{f}^{\prime}\) lies in \(E^{1},\) complex if \(D_{f}^{\prime} \subseteq C,\) vector valued if \(D_{f}^{\prime}\) is a subset of \(E^{n},\) and scalar valued if \(D_{f}^{\prime}\) lies in the scalar field of \(E^{n} .(\text { 'ln the latter two cases, we use the same terminology if }\) \(E^{n}\) is replaced by some other (fixed) normed space under consideration.) The domain \(A\) may be arbitrary.
For such functions one can define various operations whenever they are defined for elements of their ranges, to which the function values \(f(x)\) belong. Thus as in Chapter 3, §9, we define the functions \(f \pm g, f g,\) and \(f / g\) "pointwise," setting
\[(f \pm g)(x)=f(x) \pm g(x), \quad(f g)(x)=f(x) g(x), \text{ and } \left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}\]
whenever the right side expressions are defined. We also define \(|f| : A \rightarrow E^{1}\) by
\[(\forall x \in A) \quad|f|(x)=|f(x)|.\]
In particular, \(f \pm g\) is defined if \(f\) and \(g\) are both vector valued or both scalar valued, and \(f g\) is defined if \(f\) is vector valued while \(g\) is scalar valued; similarly for \(f / g .\) (However, the domain of \(f / g\) consists of those \(x \in A\) only for which \(g(x) \neq 0 . )\)
In the theorems below, all limits are at some (arbitrary, but fixed) point \(p\) of the domain space \((S, \rho) .\) For brevity, we often omit \(" x \rightarrow p."\)
For any functions \(f, g, h : A \rightarrow E^{1}(C), A \subseteq(S, \rho),\) we have the following:
- (i) If \(f, g, h\) are continuous at \(p(p \in A),\) so are \(f \pm g\) and fh. So also is \(f / h,\) provided \(h(p) \neq 0 ;\) similarly for relative continuity over \(B \subseteq A\).
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(ii) If \(f(x) \rightarrow q, g(x) \rightarrow r,\) and \(h(x) \rightarrow a(\text { all, as } x \rightarrow p \text { over } B \subseteq A),\) then
- \(f(x) \pm g(x) \rightarrow q \pm r\)
- \(f(x) h(x) \rightarrow q a ;\) and
- \(\frac{f(x)}{h(x)} \rightarrow \frac{q}{a},\) provided \(a \neq 0\)
All this holds also if \(f\) and \(g\) are vector valued and \(h\) is scalar valued.
For a simple proof, one can use Theorem 1 of Chapter 3, §15. (An independent proof is sketched in Problems 1-7 below.)
We can also use the sequential criterion (Theorem 1 in §2). To prove (ii), take any sequence
\[\left\{x_{m}\right\} \subseteq B-\{p\}, x_{m} \rightarrow p\]
Then by the assumptions made,
\[ f\left(x_{m}\right) \rightarrow q, g\left(x_{m}\right) \rightarrow r, \text{ and } h\left(x_{m}\right) \rightarrow a \]
Thus by Theorem 1 of Chapter 3, §15,
\[f\left(x_{m}\right) \pm g\left(x_{m}\right) \rightarrow q \pm r, f\left(x_{m}\right) g\left(x_{m}\right) \rightarrow q a, \text{ and } \frac{f\left(x_{m}\right)}{g\left(x_{m}\right)} \rightarrow \frac{q}{a}\]
As this holds for any sequence \(\left\{x_{m}\right\} \subseteq B-\{p\}\) with \(x_{m} \rightarrow p,\) our assertion (ii) follows by the sequential criterion; similarly for (i).
Note 1. By induction, the theorem also holds for sums and products of any finite number of functions (whenever such products are defined).
Note 2. Part (ii) does not apply to infinite limits \(q, r, a ;\) but it does apply to limits at \(p=\pm \infty\) (take \(E^{*}\) with a suitable metric for the space \(S )\).
Note 3. The assumption \(h(x) \rightarrow a \neq 0(\text { as } x \rightarrow p \text { over } B)\) implies that \(h(x) \neq 0\) for \(x\) in \(B \cap G_{\neg p}(\delta)\) for some \(\delta>0 ;\) see Problem 5 below. Thus the quotient function \(f / h\) is defined on \(B \cap G_{\neg p}(\delta)\) at least.
II. If the range space of \(f\) is \(E^{n}\left(^{*} \text { or } C^{n}\right),\) then each function value \(f(x)\) is a vector in that space; thus \(n\) real (* respectively, complex) components, denoted
\[f_{k}(x), \quad k=1,2, \ldots, n.\]
Here we may treat \(f_{k}\) as a mapping of \(A=D_{f}\) into \(E^{1}(* \text { or } C) ;\) it carries each point \(x \in A\) into \(f_{k}(x),\) the \(k\) th component of \(f(x) .\) In this manner, each function
\[f : A \rightarrow E^{n}\left(^{*} C^{n}\right)\]
uniquely determines \(n\) scalar-valued maps
\[f_{k} : A \rightarrow E^{1}(C)\]
called the components of \(f .\) Notation: \(f=\left(f_{1}, \ldots, f_{n}\right)\).
Conversely, given \(n\) arbitrary functions
\[f_{k} : A \rightarrow E^{1}(C), \quad k=1,2, \ldots, n,\]
one can define \(f : A \rightarrow E^{n}\left(^{*} C^{n}\right)\) by setting
\[f(x)=\left(f_{1}(x), f_{2}(x), \ldots, f_{n}(x)\right).\]
Then obviously \(f=\left(f_{1}, f_{2}, \ldots, f_{n}\right) .\) Thus the \(f_{k}\) in turn determine \(f\) uniquely. To define a function \(f : A \rightarrow E^{n}\left(^{*} C^{n}\right)\) means to give its n components \(f_{k} .\) Note that
\[f(x)=\left(f_{1}(x), \ldots, f_{n}(x)\right)=\sum_{k=1}^{n} \overline{e}_{k} f_{k}(x), \quad \text{ i.e., } f=\sum_{k=1}^{n} \overline{e}_{k} f_{k}\]
where the \(\overline{e}_{k}\) are the \(n\) basic unit vectors; see Chapter 3, §1-3, Theorem 2. Our next theorem shows that the limits and continuity of \(f\) reduce to those of the \(f_{k} .\)
(componentwise continuity and limits). For any function \(f : A \rightarrow E^{n}\left(* C^{n}\right),\) with \(A \subseteq(S, \rho)\) and with \(f=\left(f_{1}, \ldots, f_{n}\right),\) we have that
(i) \(f\) is continuous at \(p(p \in A)\) iff all its components \(f_{k}\) are, and
(ii) \(f(x) \rightarrow \overline{q}\) as \(x \rightarrow p(p \in S)\) iff
\[f_{k}(x) \rightarrow q_{k} \text{ as } x \rightarrow p \quad(k=1,2, \ldots, n),\]
i.e., iff each \(f_{k}\) has, as its limit at \(p,\) the corresponding component of \(\overline{q} .\)
Similar results hold for relative continuity and limits over a path \(B \subseteq A\).
We prove (ii). If \(f(x) \rightarrow \overline{q}\) as \(x \rightarrow p\) then, by definition,
\[(\forall \varepsilon>0)(\exists \delta>0)\left(\forall x \in A \cap G_{\neg p}(\delta)\right) \quad \varepsilon>|f(x)-\overline{q}|=\sqrt{\sum_{k=1}^{n}\left|f_{k}(x)-q_{k}\right|^{2}};\]
in turn, the right-hand side of the inequality given above is no less than each
\[\left|f_{k}(x)-q_{k}\right|, \quad k=1,2, \ldots, n.\]
Thus
\[(\forall \varepsilon>0)(\exists \delta>0)\left(\forall x \in A \cap G_{\neg p}(\delta)\right) \quad\left|f_{k}(x)-q_{k}\right|<\varepsilon;\]
i.e., \(f_{k}(x) \rightarrow q_{k}, k=1, \ldots, n.\)
Conversely, if each \(f_{k}(x) \rightarrow q_{k},\) then Theorem 1 (ii) yields
\[\sum_{k=1}^{n} \overline{e}_{k} f_{k}(x) \rightarrow \sum_{k=1}^{n} \overline{e}_{k} q_{k}.\]
By formula \((1),\) then, \(f(x) \rightarrow \overline{q}\) (for \(\sum_{k=1}^{n} \overline{e}_{k} q_{k}=\overline{q} ) .\) Thus (ii) is proved; similarly for (i) and for relative limits and continuity.
Note 4. Again, Theorem 2 holds also for \(p=\pm \infty\) (but not for infinite \(q )\).
Note 5. A complex function \(f : A \rightarrow C\) may be treated as \(f : A \rightarrow E^{2}\). Thus it has two real components: \(f=\left(f_{1}, f_{2}\right) .\) Traditionally, \(f_{1}\) and \(f_{2}\) are called the real and imaginary parts of \(f,\) also denoted by \(f_{\text { re }}\) and \(f_{\text { im }},\) so
\[f=f_{\mathrm{re}}+i \cdot f_{\mathrm{im}}.\]
By Theorem \(2, f\) is continuous at \(p\) iff \(f_{\text { re }}\) and \(f_{\text { im }}\) are.
The complex exponential is the function \(f : E^{1} \rightarrow C\) defined by
\[f(x)=\cos x+i \cdot \sin x, \text{ also written } f(x)=e^{x i}.\]
As we shall see later, the sine and the cosine functions are continuous. Hence so is \(f\) by Theorem \(2 .\)
III. Next, consider functions whose domain is a set in \(E^{n}\left(^{*} \text { or } C^{n}\right) .\) We call them functions of \(n\) real \(\left(* \text { or complex) variables, treating } \overline{x}=\left(x_{1}, \ldots, x_{n}\right) \text { as }\right.\) a variable \(n\)-tuple. The range space may be arbitrary.
In particular, a monomial in \(n\) variables is a map on \(E^{n}\left(^{*} \text { or } C^{n}\right)\) given by a formula of the form
\[f(\overline{x})=a x_{1}^{m_{1}} x_{2}^{m_{2}} \cdots x_{n}^{m_{n}}=a \cdot \prod_{k=1}^{n} x_{k}^{m_{k}},\]
where the \(m_{k}\) are fixed integers \(\geq 0\) and \(a \in E^{1}\left(^{*} \text { or } a \in C\right) .^{2}\) If \(a \neq 0\), the
\(\operatorname{sum} m=\sum_{k=1}^{n} m_{k}\) is called the degree of the monomial. Thus
\[f(x, y, z)=3 x^{2} y z^{3}=3 x^{2} y^{1} z^{3}\]
defines a monomial of degree \(6,\) in three real (or complex) variables \(x, y, z\). (We often write \(x, y, z\) for \(x_{1}, x_{2}, x_{3} . )\)
A polynomial is any sum of a finite number of monomials; its degree is, by definition, that of its leading term, i.e., the one of highest degree. (There may be several such terms, of equal degree.) For example,
\[f(x, y, z)=3 x^{2} y z^{3}-2 x y^{7}\]
defines a polynomial of degree 8 in \(x, y, z .\) Polynomials of degree 1 are sometimes called linear.
A rational function is the quotient \(f / g\) of two polynomials \(f\) and \(g\) on \(E^{n}\) \(\left(^{*} \mathrm{or} C^{n}\right)\). Its domain consists of those points at which \(g\) does not vanish. For example,
\[h(x, y)=\frac{x^{2}-3 x y}{x y-1}\]
defines a rational function on points \((x, y),\) with \(x y \neq 1 .\) Polynomials and monomials are rational functions with denominator \(1 .\)
Any rational function (in particular, every polynomial) in one or several variables is continuous on all of its domain.
- Proof
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Consider first a monomial of the form
\[f(\overline{x})=x_{k} \quad(k \text{ fixed });\]
it is called the \(k\) th projection map because it "projects" each \(\overline{x} \in E^{n}\left(^{*} C^{n}\right)\) onto its \(k\) th component \(x_{k}\).
Given any \(\varepsilon>0\) and \(\overline{p},\) choose \(\delta=\varepsilon .\) Then
\[\left(\forall \overline{x} \in G_{\overline{p}}(\delta)\right) \quad|f(\overline{x})-f(\overline{p})|=\left|x_{k}-p_{k}\right| \leq \sqrt{\sum_{i=1}^{n}\left|x_{i}-p_{i}\right|^{2}}=\rho(\overline{x}, \overline{p})<\varepsilon.\]
Hence by definition, \(f\) is continuous at each \(\overline{p} .\) Thus the theorem holds for projection maps.
However, any other monomial, given by
\[f(\overline{x})=a x_{1}^{m_{1}} x_{2}^{m_{2}} \cdots x_{n}^{m_{n}},\]
is the product of finitely many (namely of \(m=m_{1}+m_{2}+\ldots+m_{n} )\) projection maps multiplied by a constant \(a\). Thus by Theorem \(1,\) it is continuous. So also is any finite sum of monomials (i.e., any polynomial), and hence so is the quotient \(f / g\) of two polynomials (i.e., any rational function) wherever it is defined, i.e., wherever the denominator does not vanish. \(\square\)
IV. For functions on \(E^{n}\left(^{*} \text { or } C^{n}\right),\) we often consider relative limits over \(a\) line of the form
\[\overline{x}=\overline{p}+t \vec{e}_{k} \text{ (parallel to the } k^{th} \text{ axis, through } \overline{p} );\]
see Chapter 3, §§4-6, Definition \(1 .\) If \(f\) is relatively continuous at \(\overline{p}\) over that line, we say that \(f\) is continuous at \(\overline{p}\) in the \(k\) th variable \(x_{k}\) (because the other components of \(\overline{x}\) remain constant, namely, equal to those of \(\overline{p},\) as \(\overline{x}\) runs over that line). As opposed to this, we say that \(f\) is continuous at \(\overline{p}\) in all \(n\) variables jointly if it is continuous at \(\overline{p}\) in the ordinary (not relative) sense. Similarly, we speak of limits in one variable, or in all of them jointly.
since ordinary continuity implies relative continuity over any path, joint continuity in all \(n\) variables always implies that in each variable separately, but the converse fails (see Problems 9 and 10 below \() ;\) similarly for limits at \(\overline{p}\).


Page 14.4: Case Study- Agriculture and the Global Bee Colony Collapse
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Two thousand years ago, at the height of the Roman Empire, the poet Virgil wrote lovingly about the practice of beekeeping, of cultivating the “aerial honey and ambrosial dews” he called “gifts of heaven” ( Georgics IV: 1-2 ). Bees represent a gift to humanity even greater that Virgil knew. In addition to satisfying the human appetite for honey, the Italian honeybee, Apis melliflora , is the world’s most active pollinator, responsible for over 80 of the world’s most common nongrain crops, including apples, berries, almonds, macadamias, pumpkins, melons, canola, avocadoes, and also coffee beans, broccoli and lettuce. Even the production chain of the enormous meat and cotton industries relies at crucial points on the ministrations of the humble honeybee. We depend on pollinated fruits, nuts and seeds for a third of our caloric intake, and for vital vitamins, minerals and antioxidants in our diet. In total, around 80% of the foods we eat are to some degree the products of bee pollination, representing one third of total agricultural output.
Given the $1 trillion value of pollinated produce, any threat to the health of honey bees represents a serious threat to the human food chain—a classic sustainability issue. With the industrialization of the global agricultural system over the last 50 years—including crop monoculture and mass fertilization—bees have indeed faced a series of threats to their ancient role, the most recent of which, so-called Colony Collapse Disorder, is the most serious yet.
In his poetic primer on beekeeping, Virgil includes a moving description of a bee colony suffering mysterious decline:
Observe the symptoms when they fall away
And languish with insensible decay.

The sick, for air, before the portal gasp,
Their feeble legs within each other clasp,
Or idle in their empty hives remain,
Benumbed with cold, and listless of their gain. (368-78
Beekeepers worldwide faced an even worse predicament in late 2006: the mysterious disappearance of entire hives of bees. Over the winter, honeybees enter a form of survival hibernation. Their populations suffer inevitable losses, but these are replenished by the Queen’s renewed laying of eggs once winter thaws. In the spring of 2007, however, hundreds of thousands of colonies in the United States did not survive the winter. A full 30% of all honeybee colonies died. Each spring since has witnessed even worse declines. Similar losses afflicted Europe and Asia. Worldwide, millions of colonies and billions of bees have perished since 2006 on account of the new bee plague.
Because the global commercial value of bee pollination is so enormous, well-funded research into colony collapse began immediately. A number of theories, some credible, some not, were quickly advanced. Several studies pointed to new or enhanced viral strains, while others suggested the toxic effect of industrial fertilization. Still others claimed that mobile phone towers were interfering with the bees’ navigations systems. Because the honeybee is a charismatic creature and features so prominently in our cultural lore—we admire their industriousness, fear their stings, call our loved ones “honey,” and talk much of Queen Bees—the story of colony collapse was quickly taken up by the media. A flurry of news stories announced CCD as an epic “disaster” and profound “mystery,” which was true in simple terms, but which cast bee decline as a new and sudden calamity for which some single culprit must be responsible.
The truth, as it is now unfolding, is more complex, and shows the importance of viewing the interactions between human and natural ecologies in systemic terms. In strictly pathogenic terms, CCD is caused by the combination of a virus (called Iridoviridae or IIV) and a microsporidian fungus called Nosema . The specific interaction between the pathogens, and why they cause bees in their millions to vacate their hives, is not understood. What is becoming clear, however, is the increasing burden being placed on bees by the human agricultural system, a burden that has rendered bees increasingly vulnerable to epidemic infection. Humans have been keeping bees for eight thousand years, and European bees were at the vanguard of the successful crop colonization of the Americas. But the numbers of bees in the United States had already declined by a third since 1950 before the arrival of CCD, owing to various viral and mite infestations, and the large scale changes in bee habitat and lifestyle.
Before the industrialization of farming, bees came from neighboring wildlands to pollinate the diverse range of crops available to them on small plots. But the conversion, for economic reasons, of arable land into enormous monocrop properties in the last sixty years, and hence the diminishment of proximate wildflower habitats, has necessitated a different system, whereby bees are trucked around the country to service one crop at a time, be it peppers in Florida, blueberries in Maine, or almonds in California. At the height of the recent almond boom, the California crop required almost the entire bee population of the United States to be fully pollinated. Wholesale suburbanization is also to blame for the destruction of the bees’ natural wildflower habitats. Be it a thousand acre cornfield or a suburban street of well-tended green lawns, to a bees’ eyes, our modern landscape, engineered to human needs, is mostly a desert.
Studies that have not identified specific culprits for CCD have nevertheless shown the extent of the long-term decline in bee health wrought by their conscription to industrial agriculture. For instance, researchers found no fewer than 170 different pesticides in samples of American honeybees, while other studies found that even bees not suffering CCD habitually carry multiple viral strains in their systems. The combined toxic and viral load for the average honeybee is enormous. In the words of Florida’s state apiarist, “I’m surprised honey bees are alive at all.” (Jacobsen, 2008, p. 137) A further study showed a decline in the immune systems of bees owing to lack of diverse nutrition. Pollinating only almonds for weeks on end, then travelling on a flatbed truck for hundreds of miles in order to service another single crop, is not the lifestyle bees have adapted to over the near 80 million years of their existence. As Virgil warned, “First, for thy bees a quiet station find.” The lives of modern bees have been anything but quiet, and the enormous changes in their habitat and lifestyle have reduced their species’ resilience.
The most important lesson of recent research into CCD is not the identification of IIV and Nosema as the specific contributors, but the larger picture it has provided of a system under multiple long-term stresses. Complex systems, such as bee pollination and colony maintenance, are not characterized by linear development, but rather by sudden, nonlinear changes of state called tipping points. CCD is an example of a potential tipping point in a natural system on which humans depend, in which sudden deterioration overtakes a population beyond its ability to rebound. Everything seems fine, until it isn’t. One day we have almonds, berries, melon, and coffee on our breakfast menu. The next day there’s a critical shortage, and we can’t afford them.
In sustainability terms, bee colony collapse is a classic “human dimensions” issue. CCD will not be “solved” simply by the development of a new anti-viral drug or pesticide targeting the specific pathogens responsible. Part of what has caused CCD is the immunosuppressive effects of generations of pesticides developed to counter previous threats to bee populations, be they microbes or mites. Our chemical intervention in the lifecycle of bees has, in evolutionary terms, “selected” for a more vulnerable bee. That is, bees’ current lack of resilience is a systemic problem in our historical relationship to bees, which dates back thousands of years, but which has altered dramatically in the last fifty years in ways that now threaten collapse. And this is to say nothing of the impact of bee colony collapse on other pollination-dependent animals and birds, which would indeed be catastrophic in biodiversity terms.
That we have adapted to bees, and they to us, is a deep cultural and historical truth, not simply a sudden “disaster” requiring the scientific solution of a “mystery.” In the light of sustainability systems analysis, the bee crisis appears entirely predictable and the problem clear cut. The difficulty arises in crafting strategies for how another complex system on a massive scale, namely global agriculture, can be reformed in order to prevent its collapse as one flow-on effect of the global crisis of the vital honey bee. The incentive for such reform could not be more powerful. The prospect of a future human diet without fruits, nuts and coffee is bleak enough for citizens of the developed world and potentially fatal for millions of others in the long term.
References
- Jacobsen, R. (2008). Fruitless Fall: The Collapse of the Honey Bee and the Coming Agricultural Crisis. New York: Bloomsbury
Review Questions
- What is the long history of the human relationship to bees, and what radical changes in that relationship have occurred over the last fifty years to bring it to the point of collapse?
- What are the implications of bee colony collapse for the global food system?
Glossary
Tipping Point
The critical moment of nonlinear change whereby a system changes suddenly from one state to another.


16.6: Eukaryotic Translational and Post-translational Gene Regulation
Skills to Develop
- Understand the process of translation and discuss its key factors
- Describe how the initiation complex controls translation
- Explain the different ways in which the post-translational control of gene expression takes place
After the RNA has been transported to the cytoplasm, it is translated into protein. Control of this process is largely dependent on the RNA molecule. As previously discussed, the stability of the RNA will have a large impact on its translation into a protein. As the stability changes, the amount of time that it is available for translation also changes.
The Initiation Complex and Translation Rate
Like transcription, translation is controlled by proteins that bind and initiate the process. In translation, the complex that assembles to start the process is referred to as the initiation complex . The first protein to bind to the RNA to initiate translation is the eukaryotic initiation factor-2 (eIF-2) . The eIF-2 protein is active when it binds to the high-energy molecule guanosine triphosphate (GTP) . GTP provides the energy to start the reaction by giving up a phosphate and becoming guanosine diphosphate (GDP) . The eIF-2 protein bound to GTP binds to the small 40S ribosomal subunit . When bound, the methionine initiator tRNA associates with the eIF-2/40S ribosome complex, bringing along with it the mRNA to be translated. At this point, when the initiator complex is assembled, the GTP is converted into GDP and energy is released. The phosphate and the eIF-2 protein are released from the complex and the large 60S ribosomal subunit binds to translate the RNA. The binding of eIF-2 to the RNA is controlled by phosphorylation. If eIF-2 is phosphorylated, it undergoes a conformational change and cannot bind to GTP. Therefore, the initiation complex cannot form properly and translation is impeded (Figure \(\PageIndex{1}\)). When eIF-2 remains unphosphorylated, it binds the RNA and actively translates the protein.
Art Connection
An increase in phosphorylation levels of eIF-2 has been observed in patients with neurodegenerative diseases such as Alzheimer’s, Parkinson’s, and Huntington’s. What impact do you think this might have on protein synthesis?
Chemical Modifications, Protein Activity, and Longevity
Proteins can be chemically modified with the addition of groups including methyl, phosphate, acetyl, and ubiquitin groups. The addition or removal of these groups from proteins regulates their activity or the length of time they exist in the cell. Sometimes these modifications can regulate where a protein is found in the cell—for example, in the nucleus, the cytoplasm, or attached to the plasma membrane.
Chemical modifications occur in response to external stimuli such as stress, the lack of nutrients, heat, or ultraviolet light exposure. These changes can alter epigenetic accessibility, transcription, mRNA stability, or translation—all resulting in changes in expression of various genes. This is an efficient way for the cell to rapidly change the levels of specific proteins in response to the environment. Because proteins are involved in every stage of gene regulation, the phosphorylation of a protein (depending on the protein that is modified) can alter accessibility to the chromosome, can alter translation (by altering transcription factor binding or function), can change nuclear shuttling (by influencing modifications to the nuclear pore complex), can alter RNA stability (by binding or not binding to the RNA to regulate its stability), can modify translation (increase or decrease), or can change post-translational modifications (add or remove phosphates or other chemical modifications).
The addition of an ubiquitin group to a protein marks that protein for degradation. Ubiquitin acts like a flag indicating that the protein lifespan is complete. These proteins are moved to the proteasome , an organelle that functions to remove proteins, to be degraded (Figure \(\PageIndex{2}\)). One way to control gene expression, therefore, is to alter the longevity of the protein.
Summary
Changing the status of the RNA or the protein itself can affect the amount of protein, the function of the protein, or how long it is found in the cell. To translate the protein, a protein initiator complex must assemble on the RNA. Modifications (such as phosphorylation) of proteins in this complex can prevent proper translation from occurring. Once a protein has been synthesized, it can be modified (phosphorylated, acetylated, methylated, or ubiquitinated). These post-translational modifications can greatly impact the stability, degradation, or function of the protein.
Art Connections
Figure \(\PageIndex{1}\): An increase in phosphorylation levels of eIF-2 has been observed in patients with neurodegenerative diseases such as Alzheimer’s, Parkinson’s, and Huntington’s. What impact do you think this might have on protein synthesis?
- Answer
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Protein synthesis would be inhibited.
Glossary
- eukaryotic initiation factor-2 (eIF-2)
- protein that binds first to an mRNA to initiate translation
- guanine diphosphate (GDP)
- molecule that is left after the energy is used to start translation
- guanine triphosphate (GTP)
- energy-providing molecule that binds to eIF-2 and is needed for translation
- initiation complex
- protein complex containing eIF2-2 that starts translation
- large 60S ribosomal subunit
- second, larger ribosomal subunit that binds to the RNA to translate it into protein
- proteasome
- organelle that degrades proteins
- small 40S ribosomal subunit
- ribosomal subunit that binds to the RNA to translate it into protein


3.1: Three Dimensional Vectors
3-Dimensional Space
To this point, we have been working with vectors in 2-dimenisional space. Now, we will expand our discussion to 3-dimensional space.
The 2-dimensional coordinate system is built around a set of two axes that intersect at right angles and one particular point called the origin. Points in the plane are described by ordered pairs \((x, y)\) and vectors in standard position by \(\langle x, y\rangle\).
The 3-dimensional coordinate system is built around a set of three axes that intersect at right angles and one particular point again called the origin. Points in the plane are described by ordered triples \((x, y, z)\) and vectors in standard position by \(\langle x, y, z\rangle\).
The Distance between two Points in 2 & 3-Dimensional Space
In two-dimensional space , the distance \(d\) between two points say \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) is given by the distance formula
\[d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} \nonumber \]
In three-dimensional space , the distance \(d\) between two points say \(P(x_1,y_1,z_1)\) and \(Q(x_2,y_2,z_2)\) is given by the distance formula
\[d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2} \nonumber \]
The distance between the two points \(P(2,\ 2,\ 5)\) and \(Q(5,\ 6,\ 2)\) is
Solution
\(\begin{gathered}
d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2} \\
d=\sqrt{(5-2)^2+(6-2)^2+(2-5)^2} \\
d=\sqrt{(3)^2+(4)^2+(-3)^2} \\
d=\sqrt{9+16+9} \\
d=\sqrt{34} \approx 5.8 \text { units }
\end{gathered}\)
The distance between the two points \(P(2,\ 2,\ 5)\) and \(Q(5,\ 6,\ 2)\) is \(\sqrt{34}\) \(\approx 5.8\\) units.
Using Technology
We can use technology to find the distance between points.
Go to www.wolframalpha.com.
To find the distance between the two points \((-3,\ 5)\) and \((-7,\ 4)\) enter distance \((-3,\ 5)\) and \((-7,\ 4)\) in the entry field. Wolframalpha tells you what it thinks you entered, then tells you its answer. In this case, \(\sqrt{17}\ \approx 4.1\) 2311.
The Equation of a Circle and a Sphere
We can use the distance formulas to get equations of circles and spheres.
The center-radius form of a circle with center at the point \(C(h,k)\) and radius \(r\) is
\[(x-h)^2+(y-k)^2=r^2 \nonumber \]
The center-radius form of a sphere with center at the point \(C(h,k,j)\) and radius \(r\) is
\[(x-h)^2+(y-k)^2+(z-j)^2=r^2 \nonumber \]
To write the equation of a circle that has the point \(C\left(4,\ 7\right)\) as its center and radius 8, we use the center-radius form \({\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2\) with \(h=4\) , \(k=7\) , and \(r=8\) .
Solution
\(\begin{aligned}
& (x-h)^2+(y-k)^2=r^2 \\
& (x-4)^2+(y-7)^2=8^2 \\
& (x-4)^2+(y-7)^2=64
\end{aligned}\)
To write the equation of a sphere that has the point \(C\left(4,\ 7,\ 1\right)\) as its center and radius 8, we use the center-radius form \({\left(x-h\right)}^2+{\left(y-k\right)}^2+\ {\left(z-j\right)}^2=r^2\) with \(h=4\) , \(k=7\) , \(j=1\) , and \(r=8\) .
Solution
\(\begin{aligned}
& (x-h)^2+(y-k)^2+(z-j)^2=r^2 \\
& (x-4)^2+(y-7)^2+(z-1)^2=8^2 \\
& (x-4)^2+(y-7)^2+(z-1)^2=64
\end{aligned}\)
Try These
Find the distance between the two points (2,4) and (-3,6). Round to one decimal place.
- Answer
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\(\sqrt{29}\ \approx 5.4\) units
Find the distance between the two points (-3,5,-6) and (7,-4,2). Round to one decimal place.
- Answer
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\(7\sqrt{5}\ \approx 15.6\) units
Write the equation of a circle that has the point C(2,9) as its center and radius 1.
- Answer
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\({\left(x-2\right)}^2+{\left(y-9\right)}^2=1\)
Write the equation of a sphere that has the point C(-2,5,-7) as its center and radius 4.
- Answer
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\({\left(x+2\right)}^2+{\left(y-5\right)}^2+\ {\left(z+7\right)}^2=16\)


5.1: Ethics
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Ethics is defined as the discipline of dealing with what is good and bad with moral duty and obligation. As students who will be entering the teaching profession in the next few years, it is imperative to understand that there is a code of ethics that you are expected to follow. Teachers are often held to higher standards than other professions, both in their classrooms and in their personal lives. Teachers are expected to be fair to all their students, not impose their personal views and not abuse their powers as educators. We will examine both the New York State and the National Educator’s Association Code of Ethics to get a better understanding of these expectations. These codes of ethics for teachers are designed to protect the rights of all students. The teaching profession requires that individuals be excellent ethical role models.
As a pre-service teacher it is also important to reflect on your own set of values and beliefs. Consider how you will interact with students, manage classroom behavior and assess students’ progress. This is just a small part of an educator’s duties. The behaviors allowable in your college years may not be acceptable as a new teacher. Everything from the way you dress and speak to how you engage in social media are closely securitized as a teacher. You need to be very conscientious of your appearance, attitudes and behaviors. Teachers are expected to have high ethical standards all the time, both in and out of the classroom.
Activity
- After reading the New York State Code of Ethics for Educators, summarize the six principles.
- Compare New York State Code of Ethics with NEA Code of Ethics. Compare and contrast the similarities and differences.
- List your own values and code of ethics in becoming a teacher.
- Define integrity and explain why it is an essential disposition for teachers.
Teachers, like other professions, are expected to follow a code of ethics to guide and inform their behavior and decision-making.
The National Education Association’s preamble for its code of ethics:
“The Educator, believing in the worth and dignity of each human being, recognizes the supreme importance of the pursuit of truth, devotion to excellence, and nurture of the democratic principles. Essential to these goals is the protection of freedom to learn and to teach and the guarantee of equal educational opportunity for all. The educator accepts the responsibility to adhere to the highest ethical standards.”
New York State Code of Ethics for Educators
Statement of Purpose
The Code of Ethics is a public statement by educators that sets clear expectations and principles to guide practice and inspire professional excellence. Educators believe a commonly held set of principles can assist in the individual exercise of professional judgment. This Code speaks to the core values of the profession. “Educator” as used throughout means all educators serving New York schools in positions requiring a certificate, including classroom teachers, school leaders and pupil personnel service providers.
Principle 1: Educators nurture the intellectual, physical, emotional, social, and civic potential of each student.
Educators promote growth in all students through the integration of intellectual, physical, emotional, social and civic learning. They respect the inherent dignity and worth of each individual. Educators help students to value their own identity, learn more about their cultural heritage, and practice social and civic responsibilities. They help students to reflect on their own learning and connect it to their life experience. They engage students in activities that encourage diverse approaches and solutions to issues, while providing a range of ways for students to demonstrate their abilities and learning. They foster the development of students who can analyze, synthesize, evaluate and communicate information effectively.
Principle 2: Educators create, support, and maintain challenging learning environments for all.
Educators apply their professional knowledge to promote student learning. They know the curriculum and utilize a range of strategies and assessments to address differences. Educators develop and implement programs based upon a strong understanding of human development and learning theory. They support a challenging learning environment. They advocate for necessary resources to teach to higher levels of learning. They establish and maintain clear standards of behavior and civility. Educators are role models, displaying the habits of mind and work necessary to develop and apply knowledge while simultaneously displaying a curiosity and enthusiasm for learning. They invite students to become active, inquisitive, and discerning individuals who reflect upon and monitor their own learning.
Principle 3: Educators commit to their own learning in order to develop their practice.
Educators recognize that professional knowledge and development are the foundations of their practice. They know their subject matter, and they understand how students learn. Educators respect the reciprocal nature of learning between educators and students. They engage in a variety of individual and collaborative learning experiences essential to develop professionally and to promote student learning. They draw on and contribute to various forms of educational research to improve their own practice.
Principle 4: Educators collaborate with colleagues and other professionals in the interest of student learning.
Educators encourage and support their colleagues to build and maintain high standards. They participate in decisions regarding curriculum, instruction and assessment designs, and they share responsibility for the governance of schools. They cooperate with community agencies in using resources and building comprehensive services in support of students. Educators respect fellow professionals and believe that all have the right to teach and learn in a professional and supportive environment. They participate in the preparation and induction of new educators and in professional development for all staff.
Principle 5: Educators collaborate with parents and community, building trust and respecting confidentiality.
Educators partner with parents and other members of the community to enhance school programs and to promote student learning. They also recognize how cultural and linguistic heritage, gender, family and community shape experience and learning. Educators respect the private nature of the special knowledge they have about students and their families and use that knowledge only in the students’ best interests. They advocate for fair opportunity for all children.
Principle 6: Educators advance the intellectual and ethical foundation of the learning community.
Educators recognize the obligations of the trust placed in them. They share the responsibility for understanding what is known, pursuing further knowledge, contributing to the generation of knowledge, and translating knowledge into comprehensible forms. They help students understand that knowledge is often complex and sometimes paradoxical. Educators are confidantes, mentors and advocates for their students’ growth and development. As models for youth and the public, they embody intellectual honesty, diplomacy, tact and fairness. (New York State Education Department, 2017).
NEA Code of Ethics
PRINCIPLE I
Commitment to the Student
The educator strives to help each student realize his or her potential as a worthy and effective member of society. The educator therefore works to stimulate the spirit of inquiry, the acquisition of knowledge and understanding, and the thoughtful formulation of worthy goals.
In fulfillment of the obligation to the student, the educator–
- Shall not unreasonably restrain the student from independent action in the pursuit of learning.
- Shall not unreasonably deny the student’s access to varying points of view.

- Shall make reasonable effort to protect the student from conditions harmful to learning or to health and safety.
- Shall not intentionally expose the student to embarrassment or disparagement.
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Shall not on the basis of race, color, creed, sex, national origin, marital status, political or religious beliefs, family, social or cultural background, or sexual orientation, unfairly–
- Exclude any student from participation in any program
- Deny benefits to any student
- Grant any advantage to any student
- Shall not use professional relationships with students for private advantage.
- Shall not disclose information about students obtained in the course of professional service unless disclosure serves a compelling professional purpose or is required by law.
PRINCIPLE II
Commitment to the Profession
The education profession is vested by the public with a trust and responsibility requiring the highest ideals of professional service.
In the belief that the quality of the services of the education profession directly influences the nation and its citizens, the educator shall exert every effort to raise professional standards, to promote a climate that encourages the exercise of professional judgment, to achieve conditions that attract persons worthy of the trust to careers in education, and to assist in preventing the practice of the profession by unqualified persons.
In fulfillment of the obligation to the profession, the educator–
- Shall not in an application for a professional position deliberately make a false statement or fail to disclose a material fact related to competency and qualifications.
- Shall not misrepresent his/her professional qualifications.
- Shall not assist any entry into the profession of a person known to be unqualified in respect to character, education, or other relevant attribute.
- Shall not knowingly make a false statement concerning the qualifications of a candidate for a professional position.
- Shall not assist a noneducator in the unauthorized practice of teaching.
- Shall not disclose information about colleagues obtained in the course of professional service unless disclosure serves a compelling professional purpose or is required by law.
- Shall not knowingly make false or malicious statements about a colleague.
- Shall not accept any gratuity, gift, or favor that might impair or appear to influence professional decisions or action.
(Adopted by the NEA 1975 Representative Assembly (National Education Association, 1975).
It is important that you look closely at the expectations this college requires of you while an education major. Professional dispositions associated with you as a pre-service teacher are noted below. Please review these dispositions and reflect on responsibilities you have as an education major. These dispositions are part of the rubric that SUNY Oneonta will evaluate you throughout the program.
SUNY ONEONTA – DIVISION OF EDUCATION
The Division of Education at SUNY Oneonta is committed to preparing educators who not only possess content knowledge but also conduct themselves professionally through the expression of appropriate professional dispositions. As defined by the Interstate Teacher Assessment and Support Consortium (InTASC), dispositions are the “habits of professional action and moral commitments that underlie the performances play a key role in how teachers do, in fact, act in practice” (Council of Chief State School Officers, 2011, p. 6).
Learner & Learning: “To ensure that each student learns new knowledge and skills, teachers must understand that learning and developmental patterns vary among individuals, that learners bring unique individual differences to the learning process, and that learners need supportive and safe learning environments to thrive” (Council of Chief State School Officers, 2011, p. 8).
The candidate :
- Demonstrates respect for cultural differences and the beliefs of others.
- Demonstrates patience and flexibility during the learning process.
- Creates a challenging learning environment that demonstrates high expectations for others.
- Develops, maintains, and models appropriate relationships within the learning environment, community, and larger diverse society.
- Demonstrates student-centered decision-making based on student needs when planning and adjusting instruction.
- Demonstrates critical thinking in written & oral form.
- Demonstrates use of Evidence-Based Practices.
- Demonstrates compliance with New York State Code of Ethics for Educators and SUNY Oneonta academic standards.
- Demonstrates initiative and responsibility for own actions: independence, going beyond what is given, seeking after knowledge and professional development, and actively seeking solutions to problems.
- Demonstrates professional demeanor and appearance appropriate to the situation.
- Fosters respectful communication among all members of the learning community.
- Is prepared for class or appointments.
- Is punctual for class or appointments.
- Demonstrates reflective practice in written or verbal form.
(SUNY Oneonta Student Teaching Handbook, 2017)
Contributors and Attributions
- Foundations of Education. Authored by : SUNY Oneonta Education Department. License : CC BY: Attribution